# The Effect of Disorder on Polymer Depinning Transitions

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## Abstract

We consider a polymer, with monomer locations modeled by the trajectory of a Markov chain, in the presence of a potential that interacts with the polymer when it visits a particular site 0. We assume that probability of an excursion of length *n* is given by \(n^{-c}\varphi(n)\) for some 1 < *c* < 2 and slowly varying \(\varphi\) . Disorder is introduced by having the interaction vary from one monomer to another, as a constant *u* plus i.i.d. mean-0 randomness. There is a critical value of *u* above which the polymer is pinned, placing a positive fraction (called the contact fraction) of its monomers at 0 with high probability. To see the effect of disorder on the depinning transition, we compare the contact fraction and free energy (as functions of *u*) to the corresponding annealed system. We show that for *c* > 3/2, at high temperature, the quenched and annealed curves differ significantly only in a very small neighborhood of the critical point—the size of this neighborhood scales as \(\beta^{1/(2c-3)}\), where *β* is the inverse temperature. For *c* < 3/2, given \(\epsilon > 0\), for sufficiently high temperature the quenched and annealed curves are within a factor of \(1-\epsilon\) for all *u* near the critical point; in particular the quenched and annealed critical points are equal. For *c* = 3/2 the regime depends on the slowly varying function \(\varphi\).

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### References

- 1.Alexander K.S. and Sidoravicius V. (2006). Pinning of polymers and interfaces by random potentials.
*Ann. Appl. Probab.*16: 636–669 MATHCrossRefMathSciNetGoogle Scholar - 2.Alexander, K.S.: Ivy on the ceiling: first-order polymer depinning transitions with quenched disorder. http://arxiv.org/list/:math.PR/0612625, (2006)
- 3.Azuma K. (1967). Weighted sums of certain dependent random variables.
*Tohoku Math. J.*19: 357–367 MATHCrossRefMathSciNetGoogle Scholar - 4.Bodineau T. and Giacomin G. (2004). On the localization transition of random copolymers near selective interfaces.
*J. Stat. Phys.*117: 801–818 MATHCrossRefMathSciNetGoogle Scholar - 5.Caravenna F., Giacomin G. and Gubinelli M. (2006). A numerical approach to copolymers at selective interfaces.
*J. Stat. Phys.*122: 799–832 MATHCrossRefMathSciNetGoogle Scholar - 6.Derrida B., Hakim V. and Vannimenus J. (1992). Effect of disorder on two-dimensional wetting.
*J. Stat. Phys.*66: 1189–1213 MATHCrossRefADSMathSciNetGoogle Scholar - 7.Doney R.A. (1997). One-sided local large deviation and renewal theorems in the case of infinite mean.
*Probab. Theory Rel. Fields*107: 451–465 MATHCrossRefMathSciNetGoogle Scholar - 8.Fisher M.E. (1984). Walks, walls, wetting and melting.
*J. Stat. Phys.*34: 667–729 MATHCrossRefADSGoogle Scholar - 9.Forgacs G., Luck J.M., Nieuwenhuizen Th. M. and Orland H. (1988). Exact critical behavior of two-dimensional wetting problems with quenched disorder.
*J. Stat. Phys.*51: 29–56 MATHCrossRefADSGoogle Scholar - 10.Galluccio S. and Graber R. (1996). Depinning transition of a directed polymer by a periodic potential: a d-dimensional solution. Phys.
*Rev. E*53: R5584–R5587 Google Scholar - 11.Giacomin G. (2007). Random Polymer Models. Cambridge University Press, Cambridge MATHGoogle Scholar
- 12.Giacomin G. and Toninelli F.L. (2006). The localized phase of disordered copolymers with adsorption.
*Alea*1: 149–180 MATHMathSciNetGoogle Scholar - 13.Giacomin G. and Toninelli F.L. (2006). Smoothing effect of quenched disorder on polymer depinning transitions.
*Commun. Math. Phys.*266: 1–16 MATHCrossRefADSMathSciNetGoogle Scholar - 14.Gotcheva V. and Teitel S. (2001). Depinning transition of a two-dimensional vortex lattice in a commensurate periodic potential.
*Phys. Rev. Lett.*86: 2126–2129 CrossRefADSGoogle Scholar - 15.Monthus C. (2000). On the localization of random heteropolymers at the interface between two selective solvents.
*Eur. Phys. J. B*13: 111–130 ADSGoogle Scholar - 16.Morita T. (1964). Statistical mechanics of quenched solid solutions with applications to diluted alloys.
*J. Math. Phys.*5: 1401–1405 CrossRefADSGoogle Scholar - 17.Mukherji S. and Bhattacharjee S.M. (1993). Directed polymers with random interaction: An exactly solvable case.
*Phys. Rev. E*48: 3483–3496 CrossRefADSMathSciNetGoogle Scholar - 18.Naidenov A. and Nechaev S. (2001). Adsorption of a random heteropolymer at a potential well revisited: location of transition point and design of sequences.
*J. Phys. A: Math. Gen.*34: 5625–5634 MATHCrossRefADSMathSciNetGoogle Scholar - 19.Nechaev S. and Zhang Y.-C. (1995). Exact solution of the 2D wetting problem in a periodic potential.
*Phys. Rev. Lett.*74: 1815–1818 CrossRefADSGoogle Scholar - 20.Nelson D.R. and Vinokur V.M. (1993). Boson localization and correlated pinning of superconducting vortex arrays.
*Phys. Rev. B*48: 13060–13097 CrossRefADSGoogle Scholar - 21.Orlandini E., Rechnitzer A. and Whittington S.G. (2002). Random copolymers and the Morita aproximation: polymer adsorption and polymer localization.
*J. Phys. A: Math. Gen.*35: 7729–7751 MATHCrossRefADSMathSciNetGoogle Scholar - 22.Seneta, E.:
*Regularly Varying Functions*. Lecture Notes in Math.**508**. Berlin: Springer-Verlag, 1976Google Scholar